Solving Trig Equations

A trigonometric equation is any equation that contains a trigonometric function of an unknown variable, such as $\sin(x) = 0.5$ or $3\tan(x) + 1 = 0$.

Unlike linear equations, trigonometric equations often have an infinite number of solutions because the functions repeat their values over regular intervals, known as the period.

Solutions are usually sought within a specific domain or interval, most commonly $0^{\circ} \le x < 360^{\circ}$ or $0 \le x < 2\pi$ radians.

The principal solution is the first value provided by a calculator using inverse functions like $\arcsin$, $\arccos$, or $\arctan$.

Sine Symmetry: The sine function is symmetrical about the $90^{\circ}$ (or $\frac{\pi}{2}$) axis within its first cycle. If $x$ is a solution to $\sin(x) = k$, then $180^{\circ} - x$ (or $\pi - x$) is also a solution.

Cosine Symmetry: The cosine function is symmetrical about the $180^{\circ}$ (or $\pi$) axis. If $x$ is a solution to $\cos(x) = k$, then $360^{\circ} - x$ (or $2\pi - x$) is another solution within the primary circle.

Tangent Periodicity: The tangent function repeats every $180^{\circ}$ (or $\pi$). If $x$ is a solution to $\tan(x) = k$, then $x + 180^{\circ}$ (or $x + \pi$) is the next solution.

These relationships allow mathematicians to find all possible angles that result in the same vertical or horizontal coordinate on the unit circle.

Step 1: Isolate the Trig Term: Rearrange the equation algebraically so that the trigonometric function is on one side and a constant is on the other (e.g., $2\cos(x) = 1 \rightarrow \cos(x) = 0.5$).

Step 2: Find the Principal Value: Use the inverse trigonometric function on a calculator to find the first angle. Ensure the calculator is in the correct mode (Degrees or Radians) as specified by the problem.

Step 3: Apply Symmetry Rules: Use the specific rule for the function (Sine, Cosine, or Tangent) to find the second solution within the $0^{\circ}$ to $360^{\circ}$ range.

Step 4: Adjust for Periodicity: If the interval is larger than one full cycle, add or subtract multiples of the period ($360^{\circ}$ for sin/cos, $180^{\circ}$ for tan) to find all valid solutions.

Degrees vs. Radians: Always check the interval units. If the interval is $0 \le x \le 2\pi$, use $\pi - x$ for sine and $2\pi - x$ for cosine instead of $180$ and $360$.

Positive vs. Negative Constants: If $\sin(x)$ is negative, the principal value from the calculator may be negative (e.g., $-30^{\circ}$). You must add $360^{\circ}$ to bring it into the standard positive range.

The 'Check' Method: Always substitute your final answers back into the original equation. If $\sin(150^{\circ})$ doesn't equal the target value in your original equation, you likely applied the wrong symmetry rule.

Sketch the Graph: Even a rough sketch of the trig wave helps visualize where the horizontal line $y=k$ intersects the curve, preventing the common mistake of missing a solution.

Boundary Awareness: Pay close attention to whether the interval uses $<$ or $\le$. For example, if the interval is $0 \le x < 360$, and your solution is $360$, it must be excluded.

Multiple Angles: If the equation is $\sin(2x) = k$, solve for $2x$ first by expanding the interval (e.g., $0 \le 2x < 720$) before dividing the final results by 2.